Atlas › Modular forms › Weight 2 › Level 853 › 853.2.a › 853.2.a.a

Newform orbit 853.2.a.a

Trace form

The 33 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$33q - 13q^{2} - 18q^{3} + 31q^{4} - 16q^{5} - q^{6} - 11q^{7} - 39q^{8} + 29q^{9} - q^{10} - 46q^{11} - 29q^{12} - 11q^{13} - 22q^{14} - 25q^{15} + 35q^{16} - 18q^{17} - 32q^{18} - 15q^{19} - 42q^{20} - 4q^{21} + 5q^{22} - 69q^{23} + 11q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 33. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{33} + 31x^{32} + 369x^{31} + 1733x^{30} - 2908x^{29} - 66587x^{28} - 210636x^{27} + 483470x^{26} + 4393101x^{25} + 4613272x^{24} - 33397421x^{23} - 93463849x^{22} + 88416285x^{21} + 632343314x^{20} + 258183257x^{19} - 2248267084x^{18} - 2609190771x^{17} + 4442902578x^{16} + 8518590998x^{15} - 4359600982x^{14} - 15142424833x^{13} + 570125987x^{12} + 15939252575x^{11} + 2843177878x^{10} - 10104526924x^{9} - 2470333050x^{8} + 3880825146x^{7} + 768868269x^{6} - 872707215x^{5} - 63076723x^{4} + 93383708x^{3} - 7237746x^{2} - 1093206x + 70811$

Atkin–Lehner signs

$p$	$w_p$
853	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = +1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = -1$ : each L-function in the orbit has odd analytic rank.

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	−13	−18	−16	−11	−46	−11	−18	−15	−69	−12	−17	−27	−18	−45	−43	−38	−66	2	−72	−61	−11	0	−105	−15	−16

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.263 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound. The label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 853]: [0, 71] (certified at ell = 2147481337)
T = [(2, 1), (3, 1)]; h of degree 70 = 1 x 70 (phi(ord eps) x dim S_k^new); 7 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(853, 2)[0]   # 853.2.a.a

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 853:2 1000